On quantum jerkum operator in quantum mechanics and its phenomenological implications in quantum field theory
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On quantum jerkum operator in quantum mechanics and its phenomenological implications in quantum field theory Rami Ahmad El-Nabulsi
Received: 17 September 2019 / Accepted: 21 January 2020 © Chapman University 2020
Abstract We introduce the notion of quantum jerkum operator in quantum mechanics based on nonlocal-in-time kinetic energy approach and discuss its implication on quantum mechanics. The presence of the jerkum operator leads to a 6th -order derivative Klein–Gordon equation which generalizes the Klein–Gordon equation obtained with the framework of Quesne–Tkachuk Lorentz-covariant deformed quantum mechanical algebra. Keywords Nonlocal-in-time kinetic energy · Quantum jerkum operator · 6th -order Klein–Gordon equation Recently, the notion of nonlocal-in-time kinetic energy introduced by Suykens in [1] has received a considerable interest in quantum mechanics and others branches of theoretical physics [2–17]. In Suykens’s classical mechanical approach, the conventional kinetic energy 21 mv 2 of a particle of mass m moving at a velocity v ≡ x(t) ˙ = d x/dt ˙ )+x(t−τ ˙ ) is simply replaced by 21 m x(t) ˙ x(t+τ , where τ is a positive constant assumed to be small relative to the 2 time scale and is connected to physical constants. This tactic is motivated from Feynman’s approach to nonrelativistic quantum mechanics where he was exploring a discrete-time numerical estimation to classical velocities in connection to a measurement process and its implications in quantum mechanics by means of the kinetic energy of the particle. More explicitly, the conventional kinetic energy is replaced by 21 xk+1ε−xk xk −xε k−1 with ε = ti+1 − ti [18], i.e., particle positions are shifted backward and forward in time. Suykens’s approach leads to several motivating features and in particular the emergence of quantization from classical standpoints. Obviously, the series expansions of x(t ± τ ) in the nonlocal kinetic energy result in the appearance of higher order derivatives (HOD) terms such as the jerk j = x (3) (the rate of change of acceleration with time) in the Lagrangian of motion which have interesting implications in quantum and classical theories. Based on Suykens’s nonlocal-in-time kinetic energy approach, a quantum acceleratum operator is constructed in [19] based on an upper limit of the acceleration, i.e., maximal acceleration a = c2 /λ (λ = 2π h¯ /mv = 2π h¯ / p represents the de-Broglie wavelength) introduced by Caianiello in his geometrical formulation of quantum mechanics [20]. Here, h¯ and c are, respectively, the Planck’s constant and the celerity of light. Although little physics is known about the maximal acceleration of the particle in quantum mechanics, arguments supporting its existence have been given in the literature [21–30]. It was observed in [20–22] that the quantum system has a Gaussian state and, as a result, the Heisenberg’s uncertainty principle xp ≥ h¯ /2 R. A. El-Nabulsi (B) Mathematics and Physics Divisions, Athens In
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